A Nearly Tight Bound for Fitting an Ellipsoid to Gaussian Random Points

We prove that for $c>0$ a sufficiently small universal constant that a random set of $c d^2/\log^4(d)$ independent Gaussian random points in $\R^d$ lie on a common ellipsoid with high probability. This nearly establishes a conjecture of \citet{SaundersonCPW12}, within logarithmic factors.The latter conjecture has attracted significant attention over the past decade, dueto its connections to machine learning and sum-of-squares lower bounds for certain statistical problems.